I Learn Mathematics from My Students
-Multiculturalism in Action
DAVID W. HENDERSON!
For twenty years I have taught a junior/senior level geometry course for mathematics majors and futrne teachers at
Cornell University The core of the course emphasizes students learning geometry with reason, understanding and
through their personal experiences of the meaning of
geometry. To accomplish this I assign a series of inviting
and challenging problems and encourage the students in
which I listen to and comment on her/his thinking The students respond to my comments and I to their responses in a
cycle that ends when both of us are satisfied The results of
this dialogue are then used to stimulate the whole class discussions The text [Henderson, 1996] is based on the
course and contains a discussion of its philosophy; in addition, the paper [Lo eta!., 1996] contains a fuller discussion
of the course and the effects on the students
What I have discovered is that tluough this process not
only have the students learned, but also I have learned
much about geometry from them At first I was surprised-How could I, an expert in geometry, learn hom
students? But this learning has continued for twenty years
and I now expect its occurrence. In fact, as I expect it more
and mme and leam to listen more effectively to them, I
find that a larger pmtion of the students in the class are
showing me something about geometry that I have never
seen before
For the past six years I have kept a record of those students who have shown me geometry that I had not seen
before I recorded each student who showed me mathematics that was new to me; if one student showed me more
than one new piece of mathematics then I only counted
that student once The results follow and cover the period
1988-1993 (I did not teach the course in 1994):
#of students
178
# who showed me
new mathematics
% who showed me
56
31%
new mathematics
In this paper I give several examples of new mathematics
(theorems and proofs) shown to me by my students. This
will be followed by some reflections on the notion of proof
which have helped me to make sense of why it is that I
learn hom my students This will lead to more data from
my students and connect the discussion to issues of multiculturalism in mathematics and to our descriptions of what
is mathematics
46
Mathematics that students have showed me
There follows a few examples of mathematics I have
learned from my students I picked these examples because
they show particularly powerful and clear proofs It is
possible that some of these new pieces of mathematics
are known by others, but at the time I had not seen them
before As fat as I know, none of these proofs has
previously appeared in print Fm conciseness and clarity,
these examples are written in my words and not in the students' words
Theorem 1: The sum of the angles of a (geodesic) triangle
on the sphere is more than a straight angle.
Here is a proof that was discovered and shown to me by
Mariah Magargee, a woman first year student who was
taking a course for "students who did not yet feel comfortable with mathematics", which is taught in the same style
and using some of the same problems as the geometry
course. In class I stressed that the students should remember that latitude circles (except for the equator) are not
geodesics and I urged them not to try to apply the notions
of parallel to latitude circles Mariah ignored my mgings
and came up with the following delightful pwof:
First note that:
Two latitude circles which are symmetric about the
equator have the property that every (great circle)
transversal has opposite interior angles congruent
This follows because the two latitudes have half-tum symmetry about any point on the equator
Now we can mimic the usual planar proof:
For the Learning of Mathematics 16,. 2 (June 1996)
FlM Publishing Association, Vancouver, British Columbia, Canada
We see that the angles of the "triangle" in the figure sum
to a straight angle. This is not a true sphericalttiangle
because the base is a segment of a latitude circle instead of
a (geodesic) great circle. If we replace this latitude segment by a great circle segment then the base angles will
increase . Clearly, then, the angles of the resulting spherical
triangle sum to more than a straight angle
You can check that any small spherical ttiangle can be
derived in this marmer
Theorem 2: Stereographic projection from a sphere to the
plane is conformal (i e . preserves the size of angles.). [If
the sphere is tangent to a plane P at its South Pole S then
stereographic projection is the projection of the sphere
from the North PoleN onto the plane P.]
The following proof was shown to me by Lucy Dladla, a
Mosotho woman from South Africa Instead of focusing on
the plane P she focused on the plane Q tangent to the
North Pole.
The angle between two great circles on the sphere is the
same as the angle between the tangent vectors to these circles at the point of their intersection x Let the tangents to
these great circles at the point x intersect the plane Q in
points I and K (if a tangent vector does not intersect the
plane then use the opposite pointing vector) Let S(x) be
the projection of x on the plane P. Now connect the points
K and L with the point N Kx KN because they are two
tangents to the sphere drawn from the same point; and
Lx IN due to the same reason D.KLx MIN by SSS,
therefore LKxL LKNL
=
=
=
=
sponds to the identity, a reflection, a rotation, a translation,
or a glide reflection This proof is experienc"d by many
students as being indirect and convoluted In particular,
Jun Kawashima, an Asian-American man, looked for a
proof that would show that each isometry was the identity,
a reflection, a rotation, a ttanslation, or a glide-reflection
more directly
The following is an outline of Jun's pwof, which also
uses the Lemma:
Iff is an isometty and MBC is any triangle, then the triangle is congruent to M 'B'C~ where f(A) ~ A; f(B) ~
B; j(C) ~ C' This congruence is either a direct congruence (no reflection needed) or not a direct congruence .
If the congruence is direct and the lines AA; BB; CC'
ar'e parallel then the isometry is a lianslation along the
direction of AA'
If the congruence is direct and the lines AA; BB; CC'
are not parallel then the isometry is a rotation about the
intersection of the perpendicular bisector of the lines
AA~ BB: CC'
If the congruence is not direct, then prolong AC and
A 'C' to two directed lines and consider the bisector, b,
of the angle between the two lines Parallel transport
this line (along one of the directed lines) to line, p, that
is equidistant from A and A~ as shown in the picture
Then we can see that the congruence (and thus by the
Lemma also f) is equal to a glide reflection along p or a
reflection through p
B
c
B'
Now the circles ar·e projected onto plane P as two curves
emerging from the point S(x), the angle between these
curves being equal to that between their tangent vectors
These tangents S(x)K' and S(x)I 'are projections of the tangents xK and xL and are therefore, intersections of the
planes NKx and NLx with the projection plane P But the
planes NKx and Nix intersect the plane Q parallel to the
plane P, along the straight lines NK and NL, therefore the
straight lines S(x)K' and S(x)I 'are parallel, respectively, to
the lines NK and NL. Thus
LK'S(x)L': LKNL: LKxL
So the question emerges: Why is it that I am learning
mathematics from my students? Clearly, a necessary condition is that I listen well to my students-if I dfd not listen
there would be no chance for me to learn from' them But
that does not answer why the students continue to come up
with mathematics that is new to me How can I make sense
of this? I find that I can make some sense of this situation
by reflecting on the notion of proof which is at the core of
understanding, thinking, and listening effectively about
mathematics
Proof as a convincing communication that
and the size of the angle is preserved
answers "Why?"
Theorem 3: Every isometry of the plane is the composition
of one, two, or three reflections and is either the identity, a
reflection, a rotation, a translation, or a glide reflection
than merely show that something is ttue Here are some
conclusions about proof that I have been led to by my
The standard proof that I knew proceeded by using the
Lemma (Every isometry is determined by its effect on any
(non-degenerate) triangle) and then showing that the effect
on a ttiangle can be accomplished by zero, one, two, or
three reflections and proving that each of these cases cone-
So, in this context, what is proof? A proof must do more
experiences as a mathematician and teacher. Many of these
understandings have come from listening to my students.
Why is 3 x 2 ~ 2 x 3? To say "It follows from the
Commutative Law", or "It can be proved from Peano's
Axioms", does not answer the why-question But most
47
people will be convinced by: "I can count three 2's and
then two 3's and see that they are both equal to the same,
six". OK, now why is 2,657,873 x 92,564 = 92,564 x
2,657,873? We canuot count this-it is too large. But is
there a way to see 3 x 2 = 2 x 3 without couuting? Yes:
0 0
oo Now rotate 90 degrees:
00
If m( a) denotes the measure of angle o; then m( a) +
m('/? = 180 degrees = m(y) + m(f3). Subtracting m(y)
from both sides we conclude that m( a) = m({3) and
thus that a is congruent to f3
This proof is most satisfying to persons for whom the
0 0
o
000
Most people will not have trouble extending this proof to
include 2,657,873 x 92,564 = 92,564 x 2,657,873, or the
more general n x m = m x n Note that for the above to
make sense I must have a meaning fOr n x m and a meaning form X nand these meanings must be different So naturally I have the question: "Why (or in what sense) are
these things related?" A proof should help me experience
relationships between the meanings-it is not just an argument to show I'HAT something is true. In my experience,
to pe1fmm the fmmal mathematical induction proof starting from Peano's Axioms does not answer anyone's whyquestions, unless it is such a question as: "Why does the
Commutative Law follow hom Peano's Axioms?" Most
people (other than logicians) seem to have little interest in
that question
Conclusion 1: In order for me to be satisfied by a proof,
the proof must answer my why-question and relate my
meanings of the concepts involved
As further evidence toward this conclusion, you have probably had the experience of reading a proof and following
each step logically but still not being satisfied because the
proof did not lead you to experience the answer to your
why-question. In fact most proofs in the literature are not
written out in such a way that it is possible to follow each
step in a logical fmmal way Even if they were so written
most proofs would be too long and complicated for a person to check each step. Furthermore, even among mathematics researchers, a formal logical proof that they can
follow step-by-step is not always satisfying. For example,
my shmtest research paper [Henderson, 1973] has a very
concise simple proof that anyone who understands the
terms involved can easily follow logically step-by-step
But, I have received more questions from other mathematicians about that paper than about any of my other resear·ch
papers and most of the questions were of the smt: "Why is
it true?", "Where did it come from?", "How did you see
it?" They accepted the proof logically, they were convinced that it was true, but were not satisfied To a large
extent I was not able to answer these questions, and I find
the same phenomenon in my students-why-questions are
hard to find and hard to articulate.
Let us look at another example-the Vertical Angle
Theorem: If l and l' are straight lines then the angle a is
congruent to the angle f3
48
The proof in most textbooks goes something like this:
meaning of congruence of angles is in terms of measure
and who see that the usual operations of arithmetic can be
applied (with some care) to measures I used to be satisfied But several years ago a student in my geometry
course objected to this proof because to her an angle is a
geometric object that is congtuent to another angle if it is
possible to rigidly move one angle until it coincides with
the other. She offered the following proof:
Let h be a half-tum rotation about the point of intersection p. Since the straight lines have half'turn symmetry about p, h( a) = f3 Thus a is congtuent to f3
My first reaction was that her argument could not possibly
be a proof-it was too simple and seemed to leave out
important parts of the standard proof But she persisted
patiently for several days and my understandings deepened. Now her proof is much more convincing to me than
the standard proof You may have had similar experiences
with mathematics
Conclusion 2: A proof that sati'ifies someone else may not
satisfy me because their meanings and why-questions are
different from mine
You may ask: "But, at least in plane geometry, isn't an
angle an angle? Don't we all agtee on what an angle is?"
Well, yes and no. Consider this acute angle:
y
/
.Aow erase parts
uf its rays:
y
Is the angle still there?
j;
How about now?
f And now?
• Andnow?
The angle is somehow at the comer. It is difficult to
express this formally. As evidence, I looked in most of the
plane geometry books in the university library and found
their definitions for "angle" I found nine different definitions! Each expressed a different meaning or aspect of
"angle" and thus, potentially, each would lead to a different proof of the Vertical Angle Theorem. For example, if
you see an angle as a pencil of rays, then reflection
through the point p will directly take angle a to angle {3.
What would be the proof if you viewed an angle as a rotation (as many "modern" books do?)
Sometimes we have legitimate why-questions even with
respect to statements traditionally accepted as axioms The
Commutative Law is one possible example . Another one is
Side-Angle-Side (or SAS): If two triangles have two sides
and the included angle of one congruent to two sides and
the included angle of the other, then the triangles are congruent. You can find SAS listed in some geometry textbooks as an axiom to be assumed, in others it is listed as a
theorem to be proved, and in still others as a definition of
two l!iangles being congruent But clea!ly, in any of these
cases one can ask, "Why is SAS tme in the plane?'' This is
especially pertinent since SAS is false for (geodesic) l!iangles on a sphere So one can natmally ask, "Why is SAS
l!ue on the plane but not on the sphere?" (Iwo sides and
the included angle determine the endpoints of the thhd side
but on the sphere there are, in general, two geodesic segments joining any two points )
Learning from students who differ fmm me
The above four examples of new mathematics are all from
students who are women or persons of color (01 both) Is
this only a coincidence? I think not As I think back over
the mathematics that I have leamed from students, the
most memorable mathematics is from students who differ
from me ( a White man) in gender or race The data also
supports this in another way
#who showed me
% who showed me
new mathematics
new mathematics
56
25
31%
men of color*
178
85
58
22
13
5
38%
all Blacks
10
4
40%
#of students
all students
White men
White women
women of color*
29%
2r
36$
5
23%
*"Persons of color" is a term used in the USA to denote persons
who are not considered White. In my class persons of color
included: Asian, Asian-American, Black African, AfricanAmerican, Native American, Hispanic, and Arab
Note that White women, men of color, and Blacks all have
higher percentages who showed me new mathematics than
the percentage of White men. This is significant because
each of these gmups is undenepresented in mathematics in
the USA2 I separated out the data for Blacks because it
stands out, pmticularly given that there me so few Blacks
participating in mathematics in the USA
I have not found similar data in the literature, though
this data is in line with [Fellows eta/, 1994, p 8] who use
a somewhat similm teaching approach and observe that:
In our experience . once young women have become
intrigued by a problem, they relate well to all aspects of
mathematical creativity-the fOrmal and systematic as well
as the intuitive For example, as often as not, the optimal
solutions to the discrete math problems we have presented-and rigorous arguments to support these solutions-are
offered first by girls
What is the explanation for this data? I do not have a clem
answer to this, but I do have a partial answer It is possible
that it is an effect of the self-selection process of the students in the course, but it also makes sense in terms of the
conclusions above.
Conclusion 3: Persons who differ the most from me !for
example, in terms of cultural background and gender) are
mo,st likely to have different meanings and thus have different why-questions and different proofs
You should check this out in your own experience. Note
that this conclusion implies that I must listen pmticularly
cmefully to the meanings and proofs expressed by persons
of color and women because there is much which they see
which I do not see. The table above shows that relatively
fewer (percentage-wise) women of color showed me new
mathematics. This may indicate that I need to listen more
carefully to women of color
We should also be more critical of the standard histories
of mathematics and mathematicians which have a decidedly Eurocenl!ic emphasis I give two examples to illusl!ate
the distottions in the histories of mathematics as it is usually presented The Pythagorean Theorem is atl!ibuted to the
Greek Pythagmas, but clear statements, uses, and understandings of the result also exist in ancient Babylon, in
ancient India, and probably in ancient China before the
time of Pythagmas Most texts atl!ibute the general solution of cubic equations to Cardano, a sixteenth century
Italian However, a careful reading of Cardano's book
shows that he thought that his technique did not wmk fm
all cubic equations and he refers the reader to the wmk of
the Arabs. In fact, Omar al'Khayyam (tenth century in
Persia) presented and proved a general (geomettic) method
which works fm finding all solutions to all cubic equations See [Henderson, 1996], Chapter 14, for more discussion of Cardano and al 'Khayyam For more examples and
discussions of the Em acentric biases in histories of mathematics see [Joseph, 1991]
Corollary: I can learn much about mathematics by listening to persons whose cultural backgrounds or gender is
different than mine Hearing someone else's proof may be
difficult and may require considerable effort and patience
on my part
Perhaps it is true that women and persons of color are
underrepresented in mathematics because they are not being
well listened to by those of us already in mathematics
People understanding and communicating
mathematics
Recently, a spirited debate smtaced in the pages of the
Bulletin of the AMS, initiated in the July 1993 issue in an
aiticle by Arthur Jaffe and Frank Quinn [Jaffe/Quinn,
1993]. It was followed in the Aprill994 issue by an article
by William P Thurston [Thmston, 1994] and by a collection of "responses" and "responses to comments" by several different mathematicians [Atiyah, 1994] Thurston
rejects the popular fmmal definltion-theorem-proof model
as an adequate description of mathematics and states that:
If what we are doing is constmcting better ways of thinking,
then psychological and social dimensions are essential to a
good model for mathematical progr.ess
The measure of our success is whether what we do
enables people to understand and think more clearly and
effectively about mathematics
49
I agree with this statement and offer this paper in that context I also offer this paper in the context of the widely
acknowledged observation that there are too few women
and persons of color participating fully in understanding
and thinking about mathematics. Research in the social sciences suggests that successful learning takes place for
many underrepresented minority and women students
when instruction builds upon personal experience and provides for a diversity of ideas and perspectives (see
[Belenky et al, 1986], [Cheek, 1984], [Valverde, 1984])
My experiences have shown me, and the data above
suggests, that as we encourage and listen more effectively
to students and, in particular, to women and persons of
color, those of us already participating in mathematics will
find that our own understandings and thinking about mathematics are being emiched. (See [Henderson, 1981].) The
kind of listening that I am talking about here and the kind
that I attempt in my classroom is described by Jere
Confrey in the context of children's mathematics:
Close listening involves an act of decentering by an adult or
possibly a peer, in order to imagine what the view of the
child might be like It includes repeated requests for a child
to explain what the problem is that she is addressing, what
she sees herself doing, and how she feels about her
progress It requires one to ask for elaboration from the
child about what, where, how, and why
A willingness to
assume a rationality in the child's responses and to relinquish or modify one's own expected responses is a necessary part of the process [Confrey, 1993]
In addition, this is also in line with recent recommendations from various national bodies, for example,
Mathematical Association of America [MAA, 1992],
National Science Foundation [NSF, 1992], the
Mathematical Sciences Education Board (MSEB) [MSEB,
1990 & MSEB, 1994] For example the MSEB recommends in mathematics education
a style of instruction that rewards exploration, encourages experiments, respects conjectural approaches to' ~olv­
ing probleni~Students need to experience genuine Problems~ those whose solutions have yet to be developed by
the students (or even perhaps by their teachers) Learning
should be guided by the search to answer questions-first at
an intuitive, empirical level; then by generalizing; and later
by justifying (proving). [MSEB, 1990, p. 14]
My teaching background
My teaching is a product of Western Civilization. My
known ancestors lived in England, Scotland, Ireland,
Germany, and Luxembourg On my mother's side I am a
descendant from a long line of academics stretching back
(according to family traditions) to at least the seventeenth
century On my father's side I am descended from working
class people who placed high value on education My
mode of teaching also has deep Western roots that reach
back to the Socratic dialogues recorded by Plato in ancient
Greece More directly, my teaching has been influenced by
my experiences in high school, college and graduate
school In Ames, Iowa, my high school world literature
teacher (Mary McNally) coaxed deep creative thinking out
50
of us through her many writing assignments At
Swarthmore College in Pennsylvania, instead of attending
classes I spent my last two years in student participation
seminars and tutorials based on an old English academic
model. At the University of Wisconsin my graduate mentor, R .H. Bing, taught without lectures or textbooks in a
style which is often known as the Moore Method, named
after Bing's own graduate mentor RL Moore at the
University of Texas (See [Taylor, 1972], for more information on the Moore Method) R.L Moore, around the
turn of the century, was one of the first Americans to eam
a Ph D in mathematics from an American university
(University of Chicago)
My teaching of the geometry course is embedded in this
background. I value diverse viewpoints in my classroom
and I try to listen to all my students and to encourage each
one to express their understandings and reasonings in their
own words It was tluough this that I eventually discovered
that I was learning from my students
Multiculturalism and our views of mathematics
As I see it, all this is part of multiculturalism. People attach
many different meanings onto "multiculturalism" and there
seems to be no widely accepted definition For my purpose
I define multiculturalism as listening to and learning from
others who come from different experiences. Some persons paint a picture of multiculturalism as a force which is
in opposition to Western Civilization and which contributes to what they perceive as a decline in Western
Civilization . From my experiences, I would say instead
that multiculturalism (in the sense I have defined) can be a
natural part of Western or of any civilization and is a force
which iS llot a cause of, but rather a significant part of the
solutiqtl to, some of the cunent problems in Western
Civilization. When we approach others with the attitude
that we can listen to ariel learn from each other then we are
all emiched
·
In my own experience I gained new views of mathematics tluough listening to my students. And these views have
also helped me to better hear the mathematics that my students are communicating Let me be cleat that I am not
advocating abandoning definitions, theorems, and proofs
in mathematics or in the teaching of mathematics In fact, I
advocate more attention to definitions and proofs at all levels of mathematics teaching But I am also advocating
looking at om assumptions, definitions, theorems, and
proofs in less formal ways, and in ways that include social
and individual aspects of experience David Hilbert, called
by some the "father of formalism," wrote in the Preface to
[Hilbert/Cohn-Vossen 1932]:
In mathematics, as in any scientific research, we fmd two
tendencies present On the one hand, the tendency toward
abstraction seeks to crystallize the logical relations inherent
in the maze of material that is being studied, and the correlate the material in a systematic and orderly manner On the
other hand, the tendency toward intuitive understanding
fosters a more immediate grasp of the objects one studies, a
live rapport with them, so to speak, which stresses the concrete meaning of their relations
As the students in my course gain a "live rappmt" with the
objects of study they construct many differing definitions
and axioms Thus the definitions and axioms come hom
theii experiences Accmding to the common formal views
of mathematics we then ~'need" to settle on one consistent
set of axioms and definitions. This is not the experience of
the students in my course. Most semesters they come up
with several different notions of "straightness", ten different notions of "angle", and seven versions of the parallel
postulate. Rather than settling on one, it seems natural to
hold and explore the complexity and interrelations among
the differing notions It is not so much that the differing
definitions and axioms contradict each other but, rather,
that they emich and supplement each other and point out
differing points of view and aspects of our live rapport
with "straightness", "angle", and "parallel". For more discussion of this issue refer to [Lo et al., 1996] and the introduction to [Henderson, 1996] There is a common notion
that it is important in mathematics to be consistent at the
level of definitions and axioms However, it is an empirically observable fact that neither mathematicians nor mathematical texts are consistent with one another For example, more than nine different definitions of angle can be
found in plane geometry texts in the Cornell Library and
there is no agreement across calculus texts as to whether
the fimctionf(x) = ifx is continuous or discontinuous. To
shackle omselves to an artificial consistency would be
foolish and limiting
But: What is the consistency of mathematics? Is it all
arbitrary? "Arbitrminess" is a dangerous notion There
may be more than one possible starting point, but that does
not mean that the starting points are arbitrary Differing
contexts, differing points of view, and differing why-questions bring with them a demand for differing starting
points. If the choice were arbitrary, then that does away
with the need for any discussion It is easy when discussing mathematics to slide from "if it is not absolute" to
"then everything is completely relative/arbitrary" In most
of these discussions there does not appear to be any middle
ground In the geometry course the class holds onto the
complexity of the multiple definitions and axioms But
each student holds onto those notions which best relate to
herfhls why-questions The course demands that each student's definitions and axioms must be meaningful in the
sense that they must lead to proofs that are convincingly
communicated to others and answer - Why?
But listening effectively is not automatic When I started
using this teaching method twenty years ago I felt threatened when I could not understand a student's questions or
explanations - after all, I was the expert in geometry.
Gradually, after much persistence from the students, I
began to realize that my old ways of understanding had
blinded me from hearing alternatives Among the old ways
that interfered with my effective listening were the common views of mathematics that are embedded in most of
our textbooks. These views emphasize precise, consistent
assumptions and definitions from which themems are
proved by applying certain logical rules to the assumptions
and definitions. According to these views, the end results
are mathematics that is certain. (See, for example, [Henley,
1991] and [Jaffe/Quinn, 1993] for arguments by mathematicians who support these types of views ) I see no way
within these views of mathematics to even start to account
for the data given in this paper. I agree with Thurston
[Thurston, 1994] that we must label these views as being
inadequate descriptions of mathematics, especially mathematical experience and progress We must proceed in
directions that include social and individual aspects of
human mathematical experiences into our descriptions of
mathematics 3 Unless we proceed in these directions, our
very descriptions of mathematics will continue to be obstacles to progress and the full participation of all peoples in
mathematics
Notes
1 I am indebted to Kelly Gaddis fOr her assistance in data gathering for
this paper and for some of the understandings expressed here which were
sharpened through my conversations with her. In addition, Chris Breen
Jane-Jane Lo, Maria Terrell, John Volmink, and the members of Jere
Confrey's mathematics education research group gave me valuable teedback on an early draft of this paper
2 It is well known that Asians as a whole are not considered to be underrepresented in the United States, but it is not well known that AsianAmericans (native-born) are vastly underrepresented at least at the PhD
level (See [NRC 1977], Table G-WF-19) In 1975, of the 15,569 persons
in the USA with a Doctorate in mathematics 14,222 were White . 121
Black, 22 Native American, 72 Hispanic, 351 foreign-bam Asian, and
only 28 native-born Asian Since 1975 the Census Bureau has not distinguished between foreign born and native born
3 In addition to the discussionS above see, for example, [Thurston, 1994],
[MSEB, 1990, 1994], [Fellows et al, 1994], and the newsletter
Humanistic Mathematics and in the mathematics education research literature, for example [Confrey, 1991]
References
[Atiyah, 1994] M Atiyah et al Responses to ·'Theoretical mathematics:
Toward a cultural synthesis of mathematics and theoretical physics",
by A Jaffe and F. Quinn, Bull. Amer. Math Soc 30, 178-207
[Belenky et al , 1986] M F Belenky, B M Clinchy, N R. Goldberger,
J M 'Yarule, Women s ways of knowing: The development of stlf,
voice. and mind New York: Basic Books
[Cheek, 1984] Increasing participation of Native Americans in mathematics, Journal for Research in Mathematics Education, 15 107-113
[Confrey, 1991] J Confrey, Learning to listen: A student's understanding
of powers of ten, in E von Glaserfeld (Ed), Radical constructivism in
mathematics education, 111-138
[Confrey, 1993] J Confrey, Learning to see children's mathematics:
Crucial challenges in constructivist reform, in K. I obin (Ed.), The
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Contributors
S.. DEKEE
Sectew d'adaptation scolaiie
Faculte d'educatlon
Universue de sheibrooke
Sherbrooke, QC, Canada J1K 2R 1
J. DIONNE, R. MURA
Faculte des sciences de 1·education
univeisite Laval
Quebec, QC, canada G 1 K 7P4
(e-mail: jean_dionne@did. ulaval ca
roberta_mura@did ulaval ca)
S. GEROFSKY
Faculty of Education
Simon Fraser University
Burnaby BC. Canada V5A 156
(e-mail gerofsky®sfu ca)
S. HAROUTUNIAN·GORDON
School of Education and Social Policy
Northwestern University
21 1 5 North Campus Drive Evanston 1L 60208 USA
(e-mail: shg@nwu edu)
D.W. HENDERSON
Department oj Mathematics
Cornell University
ithaca, NY 14853-7901. USA
(e-mail: dwh®comell edu)
DP HEWITT
School of Education
Unlversity of Birmingham
Edgbaston. Birmingham B 1 5 2 TT UK
(e-mail d p hewitt@bham ac uk)
DS TARTAKOFF
Department of Mathematics mtc 249
University of Illinois at Chicago
851 south Morgan street Chicago, /L 60607 USA
(e-mail dst@uic edu)
RS.D. THOMAS
St. John's College and Department of Applied Mathematics
University oj Manitoba
Winnipeg. MB, Canada R3T 2N2
(e-mail: [email protected] ca)
52